Theorem frege95

Description: Looking one past a pair related by transitive closure of a relation. Proposition 95 of Frege1879 p. 70. (Contributed by RP, 2-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege95.x	$$⊢X\in U$$
	frege95.y	$$⊢Y\in V$$
	frege95.z	$$⊢Z\in W$$
	frege95.r	$$⊢R\in A$$
Assertion	frege95	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to X\mathrm{t+}\left(R\right)Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege95.x	$$⊢X\in U$$
2		frege95.y	$$⊢Y\in V$$
3		frege95.z	$$⊢Z\in W$$
4		frege95.r	$$⊢R\in A$$
5		vex	$$⊢f\in \mathrm{V}$$
6	1 2 3 4 5	frege88	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \left(\forall w\left(XRw\to w\in f\right)\to \left(Rhereditaryf\to Z\in f\right)\right)\right)$$
7	6	alrimdv	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \forall f\left(\forall w\left(XRw\to w\in f\right)\to \left(Rhereditaryf\to Z\in f\right)\right)\right)$$
8	1 3 4	frege94	$$⊢\left(YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to \forall f\left(\forall w\left(XRw\to w\in f\right)\to \left(Rhereditaryf\to Z\in f\right)\right)\right)\right)\to \left(YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to X\mathrm{t+}\left(R\right)Z\right)\right)$$
9	7 8	ax-mp	$$⊢YRZ\to \left(X\mathrm{t+}\left(R\right)Y\to X\mathrm{t+}\left(R\right)Z\right)$$